| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies the formal definitions of computability across equivalent models (Turing machines, μ-recursive functions, λ-calculus, register machines, Post machines). Includes the Church–Turing thesis, partial computable functions, effective procedures, and mechanistic characterizations of algorithms. Excludes informal or intuitive notions of “effective method” unless formalized. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at symbolic and algorithmic scales: step-by-step state transitions, recursion/iteration depth, tape/register configurations, combinatory reductions, and abstract machine transitions. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Turing machines, tapes, states, transition functions, λ-terms, combinators, μ-recursive function clauses, register contents, partial computable functions, oracles, encodings of data, Gödel numbers. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Computability, partiality, halting, divergence, function totality, reduction behavior, recursion depth, normalization, determinism or nondeterminism, effective enumerability. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Machine-based models (Turing machines, register machines), function-based models (primitive recursive, μ-recursive), term-rewriting models (λ-calculus, combinatory logic), oracle-augmented models, uniform vs. non-uniform models, deterministic vs. nondeterministic models. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Current machine state, tape head position, tape contents, register values, recursion/iteration counters, λ-term reduction state, oracle query state, step count, encoding indices, current partial output. |
| | Parameterization | How variables encode and represent the system’s state. | Described through Gödel encodings, machine descriptions (transition tables), recursion schemata, λ-term syntactic structure, register-update instructions, step-by-step operational semantics, and oracle-access parameters. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Idealized infinite tapes, perfect deterministic transitions, ignoring physical resource limits, abstracting data representations, treating reductions as atomic, representing functions via canonical schemata (composition, primitive recursion, minimization). |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down when physical constraints matter (finite memory), when non-standard computational paradigms are introduced (quantum, analog), when reductions require infinite parallelism, or when recursion assumptions fail for partial functions. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Computation proceeds in discrete steps; machine operations are effectively describable; recursion schemata capture all computable functions; operational rules define full computation behavior; symbolic manipulation is sufficient to define algorithms. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes Church–Turing thesis (all reasonable models are equivalent), well-foundedness of recursive definitions, determinacy of step transitions, meaningfulness of infinite but countable computational resources, and correctness of encoding schemes. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Machine models must align with recursive-function and λ-calculus definitions; reductions between models must preserve computability; equivalence proofs must not contradict known partial/total function boundaries. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony among machine models, recursion-theoretic formalisms, λ-calculus reductions, and oracular extensions; simulation relations must be coherent; Gödel encodings must integrate smoothly with operational semantics. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Machine configurations (state, tape contents, head position), sequences of reductions in λ-calculus, recursion unfolding steps, halting vs. non-halting behavior, enumeration traces of partial computable functions, oracle query patterns, divergence patterns, step counts. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by undecidability (e.g., halting problem), inability to observe infinite computations, constraints on detecting divergence, limits of Gödel encodings for higher-type objects, and inability to finitely inspect infinite-state behaviors. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Number of computation steps, recursion depth, number of reductions, tape movement count, register updates, size of encoding, oracle call count, quantifier alternation depth (in definability analyses). |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Turing machine simulators, λ-calculus reduction engines, recursive-function evaluators, operational-semantic interpreters, oracle-machine emulators, model checkers for simple automata, program analyzers implementing recursion schemata. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Computable function defined via machine execution; partiality defined via divergence; halting defined by reaching a terminal state; reduction defined via syntactic rewrite; effective enumerability defined via step-by-step enumeration procedure. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Running Turing machine simulations, performing β-reductions, expanding μ-recursive definitions, evaluating minimization procedures, executing oracle steps, tracing enumeration procedures, measuring step-count growth. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Canonical simulation runs on benchmark functions, controlled recursion unfolding, structured λ-reduction strategies (e.g., normal order vs. applicative order), principal enumeration protocols, fixed oracle access patterns, consistent Gödel encoding across analyses. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Selecting representative computable functions (primitive recursive, partial recursive), sampling across recursion schemata, selecting λ-terms of varying complexity, sampling machine descriptions, and examining diverse halting/diverging runs. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Machine traces, tape snapshots, register-update logs, λ-reduction sequences, recursion-expansion trees, enumeration logs, oracle query logs, step-count time series, encoded functions/indices. |
| | Resolution | The granularity or precision with which data is captured. | Determined by granularity of state capture, detail of reduction logs, precision of step-count tracking, fidelity of encoding, and clarity of recursion unfolding; limited by inability to finitely record infinite computations. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying simulator correctness, checking reduction-engine consistency, validating recursion interpreters, cross-checking oracle-call behavior, confirming soundness of encoding schemes, replicating machine traces across independent systems. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Incorrect transition simulation, reduction-rule misapplication, recursion mis-expansion, encoding errors, misdetected halting behavior, oracle-call inconsistencies, divergence misclassification, or logging inaccuracies. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Equivalence of computability across models (Turing ↔ λ-calculus ↔ μ-recursive ↔ register machines); closure properties of computable functions; composition and recursion laws; minimization-induced partiality; reduction relations determining normalization paths. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Computability as invariant across encoding changes; invariance under simulation between machine models; Church–Turing invariance; substitution and reduction consistencies in λ-calculus; invariance of partial function domains across recursive schemata. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | State-transition mechanisms in Turing machines; β-reduction mechanisms in λ-calculus; recursion-generation mechanisms (composition, primitive recursion, μ-operator); encoding–decoding transformations; oracle response mechanisms in extended models. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Turing transition chain → halting or divergence; λ-term reduction path → normal form or infinite reduction; recursion unfolding → base case or minimization divergence; enumeration path → partial output sequences; oracle query paths → relative computation. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Computability, partiality, totality, reduction, normalization, divergence, simulation, encoding, effective procedure, primitive recursion, μ-minimization, universality, oracle computation. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Machine-based vs. function-based models; deterministic vs. nondeterministic models; uniform vs. non-uniform frameworks; primitive recursive vs. partial recursive functions; typed vs. untyped λ-calculus; oracle hierarchies; normal vs. applicative reduction strategies. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | β-reduction equations, recursion equations, minimization equations, state-transition tables, encoding/decoding bijections, substitution equations, fixed-point equations (e.g., Y combinator). |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Turing machines, register machines, λ-calculus reduction models, μ-recursive schemata, oracle machines, combinatory logic frameworks, abstract state-transition systems, Gödel-number encodings of computations. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Single-tape Turing machines, canonical normal-order λ-reduction, minimal recursion schemata (composition + primitive recursion + minimization), simple register machines, normalized machine encodings, finite-alphabet abstraction. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Failures when considering physical constraints (finite memory), quantum or analog computation models, infinite parallelism, higher-type computation, non-well-founded encodings, or models requiring unbounded nondeterminism. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Church–Turing thesis; universality frameworks; simulation theorems between models; recursion-theoretic unification with λ-calculus; oracle hierarchies linking machine and semantic viewpoints; fixed-point theorems linking computation and logic. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Connections to programming language semantics (λ-calculus), logic (arithmetization, Gödel coding), complexity theory, automata theory, computable analysis, formal verification, and philosophy of computation. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating machine descriptions, varying transition functions, changing evaluation strategies in λ-calculus (normal vs. applicative order), modifying recursion schemata, altering oracle availability, adjusting encoding schemes, and analyzing how these changes affect computability or divergence. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing raw computation traces without intervention: tracking reduction sequences, recursion unfolding, enumeration behavior, tape/register evolution, divergence detection, oracle query frequency, and termination patterns across different models. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing equivalence of computational models, verifying computability of specific functions under different encodings, checking simulation correctness (e.g., λ-calculus simulating Turing machines), validating recursion schemata, and testing whether functions are partial or total. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Re-running simulations across independent interpreters, repeating λ-reduction sequences, re-evaluating recursive function computations, validating oracle-machine behavior, replicating minimization outcomes, and verifying consistency of encodings under multiple implementations. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing step-count distributions, estimating divergence likelihood under random inputs, measuring normalization lengths, comparing recursion expansion depth, evaluating encoding complexity, and assessing performance variation across models. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing Turing machines, μ-recursive function schemata, λ-calculus reduction systems, and register machines by expressive power, simulation efficiency, encoding simplicity, recursion depth, determinism vs. nondeterminism, and clarity of operational semantics. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying simulation errors, misapplied reductions, incorrect recursion expansions, encoding faults, misdetected halting/diverging runs, flawed oracle responses, and inconsistencies in tape/register updates. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding encoding-specific artifacts, controlling for reduction-strategy bias, neutralizing machine-description choices, standardizing input encoding, and ensuring models are compared fairly by normalizing representation details. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing simulation correctness proofs, auditing reduction rules, verifying recursion schemata, cross-checking oracle formalizations, examining encoding proofs, and evaluating equivalence claims among computational models. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating recursion definitions, refining machine models, modifying λ-reduction schemes, redefining encodings, improving oracle frameworks, repairing inconsistencies in computability proofs, and adjusting simulation arguments. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of encoding schemes, reduction strategies, recursion definitions, oracle specifications, machine descriptions, step-trace logs, and simulation proofs; explicit articulation of limitations and assumptions. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Ensuring correctness of formal arguments, avoiding hidden assumptions in encodings or reduction schemes, maintaining reproducibility of simulations, clearly stating undecidability limitations, and responsibly communicating the boundaries of computability claims. |